On the mathematical treatment of time reversal

We present a mathematical treatment of time reversal. Two mathematical models describing approximately the propagation of the time-reversed field are proposed and discussed. Zero initial conditions are exploited in the first model, whereas the method of quasi-reversibility is adopted when constructing the second model. Since computer simulation of time reversal requires knowledge of material properties of a propagating medium, such as the sound speed or electrical conductivity, the general problem of time reversal is nonlinear and ill posed. The ill-posedness is due to the nonuniqueness and instability. To treat this problem, a two-stage procedure is proposed and justified. In the first stage, the unknown material properties of a propagating inhomogeneous medium are approximately determined. Since weak scattering is not assumed, the convexification approach is adopted to estimate such properties. In the second stage, the time-reversed field is approximately determined from the solution of the Cauchy problem for a hyperbolic equation with the lateral data by the method of quasi-reversibility.

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